采用分形理论与微分几何的摆线针轮形貌模型
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摘要
根据分形理论与微分几何,应用Weierstrass-Mandelbrot函数与矢量函数构建了摆线轮齿与针齿的表面形貌模型,该模型综合体现了摆线轮齿与针齿的宏观特征与微观特征;应用MATLAB绘制了各向同性粗糙针齿及单个摆线轮齿的二维截面图,构建了摆线轮齿与针齿的接触比例系数。通过分析可知:接触比例系数随啮合点的变化而变化,且针齿与摆线轮齿内凹部分接触时的接触比例系数远大于针齿与摆线轮齿外凸部分接触时的值;接触比例系数随分形维数的增加而减小,随摆线轮与针轮中心距的增加先增大后减小,随特征尺度、针齿半径和针轮齿数的增加而增大。
The key point to analyze the contact characteristics of cycloid and pin wheels is to establish a contact model with comprehensive embodiment of the gross feature and microcharacteristic.A surface topography model of rough surfaces is established based on WeierstrassMandelbrot function and differential geometry.The function of cycloid is determined by using a vector function with modification being taken into account.The unit normal vector is calculated with the application of differential geometry.The 2 Dsections of a pin tooth and a cycloid tooth which is assumed as isotropic are plotted by MATLAB.The contact model between cylinder and rigid flat is established,aparameter called equivalent contact angle is proposed,and the relations of the angle with fractal dimension,characteristic scale and radius of cylinders are studied.Furthermore,the contact proportionality coefficient that is used to describe the contact area between cycloid and pin tooth surfaces is given.The coefficient is less than 1 in any case,and the value for the area between pin tooth and inter part of cycloid is much larger than that for the area in outer part of cycloid.It is indicated by analysis that the value of the coefficient decreases with the increase of the fractal dimension,increases at first and then decreases with the increase of the center distance,and increases with the increase of pin tooth radius,characteristic scale,and number of pin teeth.
The key point to analyze the contact characteristics of cycloid and pin wheels is to establish a contact model with comprehensive embodiment of the gross feature and microcharacteristic.A surface topography model of rough surfaces is established based on WeierstrassMandelbrot function and differential geometry.The function of cycloid is determined by using a vector function with modification being taken into account.The unit normal vector is calculated with the application of differential geometry.The 2 Dsections of a pin tooth and a cycloid tooth which is assumed as isotropic are plotted by MATLAB.The contact model between cylinder and rigid flat is established,aparameter called equivalent contact angle is proposed,and the relations of the angle with fractal dimension,characteristic scale and radius of cylinders are studied.Furthermore,the contact proportionality coefficient that is used to describe the contact area between cycloid and pin tooth surfaces is given.The coefficient is less than 1 in any case,and the value for the area between pin tooth and inter part of cycloid is much larger than that for the area in outer part of cycloid.It is indicated by analysis that the value of the coefficient decreases with the increase of the fractal dimension,increases at first and then decreases with the increase of the center distance,and increases with the increase of pin tooth radius,characteristic scale,and number of pin teeth.
引文
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[17]HUANG Kang,XIONG Yangshou,WANG Tao.Research on the dynamic response of high-contact-ratio spur gears influenced by surface roughness under EHL condition[J].Applied Surface Science,2017,392:8-18.
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[19]BERRY M V,LEWIS Z V.On the Weierstrass-Mandelbrot fractal function[J].Proceedings of The Royal Society,1980,370:459-471.
[20]CHEN Qi,ZHAO Yucai,MA Yunbo,et al.Testing experiment of characteristic-scale coefficient about cylinder surface[J].Journal of Mechanical Strength,2014,36(5):687-697.
[21]葛世荣,朱华.摩擦学的分形[M].北京:机械工业出版社,2005:122.
[22]LITVIN F L,FUENTES A.Gear geometry and applied theory[M].New York,USA:Cambridge University Press,2004:59.
[23]田红亮,陈甜敏,郑金华,等.平行轴圆柱副接触分析[J].西安交通大学学报,2016,50(1):1-8.TIAN Hongliang,CHEN Tianmin,ZHENG Jinhua,et al.Contact analysis of cylindrical pair with parallel axes[J].Journal of Xi’an Jiaotong University,2016,50(1):1-8.
[2]韩林山,柯超,齐彩娟.基于MATLAB的2K-V型减速器摆线轮疲劳强度可靠性分析[J].矿山机械,2013,41(4):131-134.HAN Linshan,KE Chao,QI Caijuan.Reliability analysis on fatigue strength of cycloid gear in 2K-V reducer based on MATLAB[J].Mining&Processing Equipment,2013,41(4):131-134.
[3]HSIEH C.The effect on dynamics of using a new transmission design for eccentric speed reducers[J].Mechanism and Machine Theory,2014,80:1-16.
[4]HSIEH C.Traditional versus improved designs for cycloidal speed reducers with a small tooth difference:the effect on dynamics[J].Mechanism and Machine Theory,2015,86:15-35.
[5]WANG Jianing,GU Jingjun,YAN Yonghua.Study on the relationship between the stiffness of RV reducer and the profile modification method of cycloid-pin wheel[C]∥KUBOTA N,KIGUCHI K,LIU Honghai,et al.Lecture Notes in Computer Science:9834Intelligent Robotics and Applications,ICIRA 2016.Berlin:Springer-Verlag,2016:721-735.
[6]LI Shuting.Design and strength analysis methods of the trochoidal gear reducers[J].Mechanism and Machine Theory,2014,81:140-154.
[7]YU Hongliu,YI Jinhua,HU Xin,et al.Study on teeth profile modification of cycloid reducer based on non-Hertz elastic[J].Mechanics Research Communications,2013,48:87-92.
[8]MANDELBORT B B.The fractal geometry of nature[M].New York,USA:W.H.Freeman and Company,1982:35-45.
[9]BROWN C A,SAVARY G.Describing ground surface texture using contact profilometry and fractal analysis[J].Wear,1991,141:211-226.
[10]MAJUMDAR A,BHUSHAN B.Role of fractal geometry in roughness characterization and contact mechanics of surfaces[J].ASME Journal of Tribology,1990,112:205-216.
[11]MAJUMDAR A,BHUSHAN B.Fractal model of elastic-plastic contact between rough surfaces[J].ASME Journal of Tribology,1991,113:1-11.
[12]MORAG Y,ETSION I.Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces[J].Wear,2007,262:624-629.
[13]MIAO Xiaomei,HUANG Xiaodiao.A complete contact model of a fractal rough surface[J].Wear,2014,309:146-151.
[14]黄康,赵韩,陈奇.两圆柱体表面接触承载能力的分形模型研究[J].摩擦学学报,2008,28(6):529-533.HUANG Kang,ZHAO Han,CHEN Qi.Research of fractal contact model on contact carrying capacity of two cylinders’surface[J].Tribology,2008,28(6):529-533.
[15]陈奇,黄康,张彦,等.基于分形接触模型的齿轮接触强度影响参数分析[J].中国机械工程,2013,16(16):2208-2211.CHEN Qi,HUANG Kang,ZHANG Yan,et al.Analysis of influence parameters on gear’s contact strength based on fractal contact model[J].China Mechanical Engineering,2013,16(16):2208-2211.
[16]CHEN Qi,XU Fan,LIU Peng,et al.Research on fractal model of normal contact stiffness between two spheroidal joint surfaces considering friction factor[J].Tribology International,2016,97:256-264.
[17]HUANG Kang,XIONG Yangshou,WANG Tao.Research on the dynamic response of high-contact-ratio spur gears influenced by surface roughness under EHL condition[J].Applied Surface Science,2017,392:8-18.
[18]陈奇,黄守武,张振,等.考虑摩擦因素的两圆柱体表面接触承载能力的分形模型研究[J].机械工程学报,2016,52(7):114-121.CHEN Qi,HUANG Shouwu,ZHANG Zhen,et al.Research on fractal contact model for contact carrying capacity of two cylinders’surfaces considering friction factors[J].Journal of Mechanical Engineering,2016,52(7):114-121.
[19]BERRY M V,LEWIS Z V.On the Weierstrass-Mandelbrot fractal function[J].Proceedings of The Royal Society,1980,370:459-471.
[20]CHEN Qi,ZHAO Yucai,MA Yunbo,et al.Testing experiment of characteristic-scale coefficient about cylinder surface[J].Journal of Mechanical Strength,2014,36(5):687-697.
[21]葛世荣,朱华.摩擦学的分形[M].北京:机械工业出版社,2005:122.
[22]LITVIN F L,FUENTES A.Gear geometry and applied theory[M].New York,USA:Cambridge University Press,2004:59.
[23]田红亮,陈甜敏,郑金华,等.平行轴圆柱副接触分析[J].西安交通大学学报,2016,50(1):1-8.TIAN Hongliang,CHEN Tianmin,ZHENG Jinhua,et al.Contact analysis of cylindrical pair with parallel axes[J].Journal of Xi’an Jiaotong University,2016,50(1):1-8.